Solving a Math Poem

Date: 05/24/2000 at 14:04:06
From: Amber
Subject: Math poem
Take five times which plus half of what
And make the square of what you've got.
Divide by one and thirty square
To get just four - that's right, it's there.
Now two more points I must impress:
Both of which and what are fractionless,
And what less which is not a lot:
Just two or three. So now, what's what?
My teacher gave me this problem to try and figure out but every time I
look at it, it doesn't make sense. I get stuck at the beginning.
Please help me!
Thank-you very much for your time and effort,
Amber

Date: 05/25/2000 at 11:24:01
From: Doctor TWE
Subject: Re: Math poem
Hi Amber - thanks for writing to Dr. Math.
When I get a problem like this, the first thing I do is read it over
several times, trying to understand a little bit more of it each time.
(I had to read this one over three times before I understood what it
was asking, and then I had to reread parts of it again as I worked it
out!)
To solve this, you need to realize that 'what' and 'where' are
numbers, which we'll represent with variables. I'll use X = what and
Y = where.
Now it helps to write the word problem (poem) in the form of an
algebraic equation. The problem is that English - especially when used
poetically - is not as precise as algebra, so some parts are open to
multiple interpretations. I'll get you started:
"Take five times which plus half of what,"
This is easy; algebraically it's: 5*Y + (1/2)X
"...and make the square of what you've got."
Here we have to decide if the word 'what' means the variable X or is
just part of the expression "what you've got." If we interpret the
"and make the square of what" as X^2, the "you've got" doesn't make
any sense. So I interpret this as meaning take the square of the
expression so far. That's: [5*Y + (1/2)X]^2
"Divide by one and thirty square,"
Again, this is not clear. This could mean divide by 31^2 = 961, or it
could mean divide by 1 + 30^2 = 901. I'll leave it to you to figure
out which interpretation is correct (only one works). Now we have
either:
[5*Y + (1/2)X]^2 / (31^2)
or [5*Y + (1/2)X]^2 / (1+30^2)
"...to get just four-that's right, it's there."
This completes the equation. We have either:
[5*Y + (1/2)X]^2 / (31^2) = 4
or [5*Y + (1/2)X]^2 / (1+30^2) = 4
"... both of which and what are fractionless,"
This means that X and Y are integers. This is useful information,
especially if you have to resort to trial-and-error, since you only
have to try integer combinations.
"...and what less which is not a lot: just two or three."
This translates to: X - Y = 2 or 3. This gives us four possible
combinations to try:
[5*Y+(1/2)X]^2 / (31^2) = 4 and X - Y = 2
[5*Y+(1/2)X]^2 / (31^2) = 4 and X - Y = 3
[5*Y+(1/2)X]^2 / (1+30^2) = 4 and X - Y = 2
[5*Y+(1/2)X]^2 / (1+30^2) = 4 and X - Y = 3
"So now, what's what?"
This tells us that they're looking for the value of X. Can you take it
from there?
I hope this helps. If you have any more questions, write back.
- Doctor TWE, The Math Forum
http://mathforum.org/dr.math/